Question: Simplify the following expression and state the condition under which the simplification is valid: $x = \dfrac{z^2 + 15z + 56}{z^2 + z - 56}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{z^2 + 15z + 56}{z^2 + z - 56} = \dfrac{(z + 7)(z + 8)}{(z - 7)(z + 8)} $ Notice that the term $(z + 8)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(z + 8)$ gives: $x = \dfrac{z + 7}{z - 7}$ Since we divided by $(z + 8)$, $z \neq -8$. $x = \dfrac{z + 7}{z - 7}; \space z \neq -8$